Orbital Stability of Periodic Traveling Waves in the $b$-Camassa-Holm Equation

TL;DR

This paper establishes criteria for the nonlinear orbital stability of periodic traveling waves in the b-family Camassa-Holm equation, applicable across a broad parameter range using Hamiltonian structure and conserved quantities.

Contribution

It provides a general stability criterion for periodic waves in the b-Camassa-Holm equation, extending beyond integrable cases using Jacobians of conserved quantities.

Findings

Stability criteria expressed via Jacobians of conserved quantities.

Applicable to all b>1, including non-integrable cases.

Identifies conditions for orbital stability of periodic traveling waves.

Abstract

In this paper, we identify criteria that guarantees the nonlinear orbital stability of a given periodic traveling wave solution within the b-family Camassa-Holm equation. These periodic waves exist as 3-parameter families (up to spatial translations) of smooth traveling wave solutions, and their stability criteria are expressed in terms of Jacobians of the conserved quantities with respect to these parameters. The stability criteria utilizes a general Hamiltonian structure which exists for every $b>1$, and hence applies outside of the completely integrable cases ($b=2$ and $b=3$).